Problem: $h'(x)=\dfrac{2h(x)}{x\ln(x)}$ Is $h(x)=4\ln(x)$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
In order to find whether $h(x)=4\ln(x)$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $h(x)$, we need to find the corresponding $h'(x)$ expression to substitute into the equation: $\begin{aligned} h'(x)&=\dfrac{d}{dx}\left[4\ln(x)\right] \\\\ &=\dfrac{4}{x} \end{aligned}$ Now we substitute ${h(x)=4\ln(x)}$ and ${h'(x)=\dfrac{4}{x}}$ into the equation: $\begin{aligned} {h'(x)}&=\dfrac{2{h(x)}}{x\ln(x)} \\\\ {\dfrac{4}{x}}&\stackrel{?}{=}\dfrac{2\left({4\ln(x)}\right)}{x\ln(x)} \\\\ \dfrac{4}{x}&\stackrel{?}{=}\dfrac{8\ln(x)}{x\ln(x)} \\\\ \dfrac{4}{x}&\stackrel{?}{=}\dfrac{8}{x} \\\\ 4&\neq 8 \end{aligned}$ We did not obtain equivalent expressions on each side. In conclusion, no, $h(x)=4\ln(x)$ is not a solution to the differential equation.